Integrand size = 24, antiderivative size = 361 \[ \int \frac {x}{216+108 x^2+324 x^3+18 x^4+x^6} \, dx=-\frac {\arctan \left (\frac {3 \sqrt [3]{-3} 2^{2/3}-2 x}{\sqrt {6 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{36 \sqrt [6]{2} 3^{5/6} \left (1+\sqrt [3]{-1}\right )^2 \sqrt {4-3 (-3)^{2/3} \sqrt [3]{2}}}+\frac {\sqrt [3]{-1} \arctan \left (\frac {3 (-2)^{2/3} \sqrt [3]{3}+2 x}{\sqrt {6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{54\ 2^{2/3} 3^{5/6} \sqrt {8+9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}}}+\frac {\text {arctanh}\left (\frac {\sqrt [6]{2} \left (3 \sqrt [3]{3}+\sqrt [3]{2} x\right )}{\sqrt {3 \left (-4+3 \sqrt [3]{2} 3^{2/3}\right )}}\right )}{108 \sqrt [6]{2} 3^{5/6} \sqrt {-4+3 \sqrt [3]{2} 3^{2/3}}}+\frac {(-1)^{2/3} \log \left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )}{216 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^2}-\frac {(-1)^{2/3} \log \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )}{648 \sqrt [3]{2} 3^{2/3}}-\frac {\log \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )}{648 \sqrt [3]{2} 3^{2/3}} \] Output:
-1/216*arctan((3*(-3)^(1/3)*2^(2/3)-2*x)/(24-18*(-3)^(2/3)*2^(1/3))^(1/2)) *2^(5/6)*3^(1/6)/(1+(-1)^(1/3))^2/(4-3*(-3)^(2/3)*2^(1/3))^(1/2)+1/324*(-1 )^(1/3)*arctan((3*(-2)^(2/3)*3^(1/3)+2*x)/(24+18*(-2)^(1/3)*3^(2/3))^(1/2) )*2^(1/3)*3^(1/6)/(8+9*I*2^(1/3)*3^(1/6)+3*2^(1/3)*3^(2/3))^(1/2)+1/648*ar ctanh(2^(1/6)*(3*3^(1/3)+2^(1/3)*x)/(-12+9*2^(1/3)*3^(2/3))^(1/2))*2^(5/6) *3^(1/6)/(-4+3*2^(1/3)*3^(2/3))^(1/2)+1/1296*(-1)^(2/3)*ln(6-3*(-3)^(1/3)* 2^(2/3)*x+x^2)*2^(2/3)*3^(1/3)/(1+(-1)^(1/3))^2-1/3888*(-1)^(2/3)*ln(6+3*( -2)^(2/3)*3^(1/3)*x+x^2)*2^(2/3)*3^(1/3)-1/3888*ln(6+3*2^(2/3)*3^(1/3)*x+x ^2)*2^(2/3)*3^(1/3)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.02 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.16 \[ \int \frac {x}{216+108 x^2+324 x^3+18 x^4+x^6} \, dx=\frac {1}{6} \text {RootSum}\left [216+108 \text {$\#$1}^2+324 \text {$\#$1}^3+18 \text {$\#$1}^4+\text {$\#$1}^6\&,\frac {\log (x-\text {$\#$1})}{36+162 \text {$\#$1}+12 \text {$\#$1}^2+\text {$\#$1}^4}\&\right ] \] Input:
Integrate[x/(216 + 108*x^2 + 324*x^3 + 18*x^4 + x^6),x]
Output:
RootSum[216 + 108*#1^2 + 324*#1^3 + 18*#1^4 + #1^6 & , Log[x - #1]/(36 + 1 62*#1 + 12*#1^2 + #1^4) & ]/6
Time = 1.50 (sec) , antiderivative size = 354, normalized size of antiderivative = 0.98, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {2466, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {x}{x^6+18 x^4+324 x^3+108 x^2+216} \, dx\) |
| \(\Big \downarrow \) 2466 |
| \(\displaystyle 1259712 \int \left (-\frac {(-1)^{2/3} \left (3 \sqrt [3]{-3} 2^{2/3}-x\right )}{136048896 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^2 \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )}-\frac {(-1)^{2/3} \left (x+3 (-2)^{2/3} \sqrt [3]{3}\right )}{408146688 \sqrt [3]{2} 3^{2/3} \left (x^2+3 (-2)^{2/3} \sqrt [3]{3} x+6\right )}-\frac {\sqrt [3]{2} x+6 \sqrt [3]{3}}{408146688\ 6^{2/3} \left (x^2+3\ 2^{2/3} \sqrt [3]{3} x+6\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 1259712 \left (\frac {\sqrt [3]{-1} \arctan \left (\frac {2 x+3 (-2)^{2/3} \sqrt [3]{3}}{\sqrt {6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{136048896 \sqrt [6]{2} 3^{5/6} \sqrt {4+3 \sqrt [3]{-2} 3^{2/3}}}-\frac {\arctan \left (\frac {\sqrt [6]{2} \left (3 \sqrt [3]{-3}-\sqrt [3]{2} x\right )}{\sqrt {3 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{45349632 \sqrt [6]{2} 3^{5/6} \left (1+\sqrt [3]{-1}\right )^2 \sqrt {4-3 (-3)^{2/3} \sqrt [3]{2}}}+\frac {\text {arctanh}\left (\frac {\sqrt [6]{2} \left (\sqrt [3]{2} x+3 \sqrt [3]{3}\right )}{\sqrt {3 \left (3 \sqrt [3]{2} 3^{2/3}-4\right )}}\right )}{136048896 \sqrt [6]{2} 3^{5/6} \sqrt {3 \sqrt [3]{2} 3^{2/3}-4}}+\frac {(-1)^{2/3} \log \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )}{272097792 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^2}-\frac {(-1)^{2/3} \log \left (x^2+3 (-2)^{2/3} \sqrt [3]{3} x+6\right )}{816293376 \sqrt [3]{2} 3^{2/3}}-\frac {\log \left (x^2+3\ 2^{2/3} \sqrt [3]{3} x+6\right )}{816293376 \sqrt [3]{2} 3^{2/3}}\right )\) |
Input:
Int[x/(216 + 108*x^2 + 324*x^3 + 18*x^4 + x^6),x]
Output:
1259712*(((-1)^(1/3)*ArcTan[(3*(-2)^(2/3)*3^(1/3) + 2*x)/Sqrt[6*(4 + 3*(-2 )^(1/3)*3^(2/3))]])/(136048896*2^(1/6)*3^(5/6)*Sqrt[4 + 3*(-2)^(1/3)*3^(2/ 3)]) - ArcTan[(2^(1/6)*(3*(-3)^(1/3) - 2^(1/3)*x))/Sqrt[3*(4 - 3*(-3)^(2/3 )*2^(1/3))]]/(45349632*2^(1/6)*3^(5/6)*(1 + (-1)^(1/3))^2*Sqrt[4 - 3*(-3)^ (2/3)*2^(1/3)]) + ArcTanh[(2^(1/6)*(3*3^(1/3) + 2^(1/3)*x))/Sqrt[3*(-4 + 3 *2^(1/3)*3^(2/3))]]/(136048896*2^(1/6)*3^(5/6)*Sqrt[-4 + 3*2^(1/3)*3^(2/3) ]) + ((-1)^(2/3)*Log[6 - 3*(-3)^(1/3)*2^(2/3)*x + x^2])/(272097792*2^(1/3) *3^(2/3)*(1 + (-1)^(1/3))^2) - ((-1)^(2/3)*Log[6 + 3*(-2)^(2/3)*3^(1/3)*x + x^2])/(816293376*2^(1/3)*3^(2/3)) - Log[6 + 3*2^(2/3)*3^(1/3)*x + x^2]/( 816293376*2^(1/3)*3^(2/3)))
Int[(u_.)*(Q6_)^(p_), x_Symbol] :> With[{a = Coeff[Q6, x, 0], b = Coeff[Q6, x, 2], c = Coeff[Q6, x, 3], d = Coeff[Q6, x, 4], e = Coeff[Q6, x, 6]}, Sim p[1/(3^(3*p)*a^(2*p)) Int[ExpandIntegrand[u*(3*a + 3*Rt[a, 3]^2*Rt[c, 3]* x + b*x^2)^p*(3*a - 3*(-1)^(1/3)*Rt[a, 3]^2*Rt[c, 3]*x + b*x^2)^p*(3*a + 3* (-1)^(2/3)*Rt[a, 3]^2*Rt[c, 3]*x + b*x^2)^p, x], x], x] /; EqQ[b^2 - 3*a*d, 0] && EqQ[b^3 - 27*a^2*e, 0]] /; ILtQ[p, 0] && PolyQ[Q6, x, 6] && EqQ[Coef f[Q6, x, 1], 0] && EqQ[Coeff[Q6, x, 5], 0] && RationalFunctionQ[u, x]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.04 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.15
| method | result | size |
| default | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )}{\sum }\frac {\textit {\_R} \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{5}+12 \textit {\_R}^{3}+162 \textit {\_R}^{2}+36 \textit {\_R}}\right )}{6}\) | \(54\) |
| risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )}{\sum }\frac {\textit {\_R} \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{5}+12 \textit {\_R}^{3}+162 \textit {\_R}^{2}+36 \textit {\_R}}\right )}{6}\) | \(54\) |
Input:
int(x/(x^6+18*x^4+324*x^3+108*x^2+216),x,method=_RETURNVERBOSE)
Output:
1/6*sum(_R/(_R^5+12*_R^3+162*_R^2+36*_R)*ln(x-_R),_R=RootOf(_Z^6+18*_Z^4+3 24*_Z^3+108*_Z^2+216))
Timed out. \[ \int \frac {x}{216+108 x^2+324 x^3+18 x^4+x^6} \, dx=\text {Timed out} \] Input:
integrate(x/(x^6+18*x^4+324*x^3+108*x^2+216),x, algorithm="fricas")
Output:
Timed out
Time = 0.12 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.17 \[ \int \frac {x}{216+108 x^2+324 x^3+18 x^4+x^6} \, dx=\operatorname {RootSum} {\left (158171241119638192128 t^{6} - 96402615118848 t^{4} + 287743415040 t^{3} - 51018336 t^{2} - 1, \left ( t \mapsto t \log {\left (\frac {65418399445721140961280 t^{5}}{415817} + \frac {2480926457425102848 t^{4}}{415817} - \frac {39451802929737984 t^{3}}{415817} + \frac {118071997444800 t^{2}}{415817} - \frac {16745884920 t}{415817} + x - \frac {268790}{415817} \right )} \right )\right )} \] Input:
integrate(x/(x**6+18*x**4+324*x**3+108*x**2+216),x)
Output:
RootSum(158171241119638192128*_t**6 - 96402615118848*_t**4 + 287743415040* _t**3 - 51018336*_t**2 - 1, Lambda(_t, _t*log(65418399445721140961280*_t** 5/415817 + 2480926457425102848*_t**4/415817 - 39451802929737984*_t**3/4158 17 + 118071997444800*_t**2/415817 - 16745884920*_t/415817 + x - 268790/415 817)))
\[ \int \frac {x}{216+108 x^2+324 x^3+18 x^4+x^6} \, dx=\int { \frac {x}{x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216} \,d x } \] Input:
integrate(x/(x^6+18*x^4+324*x^3+108*x^2+216),x, algorithm="maxima")
Output:
integrate(x/(x^6 + 18*x^4 + 324*x^3 + 108*x^2 + 216), x)
\[ \int \frac {x}{216+108 x^2+324 x^3+18 x^4+x^6} \, dx=\int { \frac {x}{x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216} \,d x } \] Input:
integrate(x/(x^6+18*x^4+324*x^3+108*x^2+216),x, algorithm="giac")
Output:
integrate(x/(x^6 + 18*x^4 + 324*x^3 + 108*x^2 + 216), x)
Time = 22.26 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.49 \[ \int \frac {x}{216+108 x^2+324 x^3+18 x^4+x^6} \, dx=\sum _{k=1}^6\ln \left (x+\mathrm {root}\left (z^6-\frac {z^4}{1640736}+\frac {235\,z^3}{129178426752}-\frac {z^2}{3100282242048}-\frac {1}{158171241119638192128},z,k\right )\,\left (216\,x+\mathrm {root}\left (z^6-\frac {z^4}{1640736}+\frac {235\,z^3}{129178426752}-\frac {z^2}{3100282242048}-\frac {1}{158171241119638192128},z,k\right )\,\left (51018336\,x-\mathrm {root}\left (z^6-\frac {z^4}{1640736}+\frac {235\,z^3}{129178426752}-\frac {z^2}{3100282242048}-\frac {1}{158171241119638192128},z,k\right )\,\left (277947894528\,x-\mathrm {root}\left (z^6-\frac {z^4}{1640736}+\frac {235\,z^3}{129178426752}-\frac {z^2}{3100282242048}-\frac {1}{158171241119638192128},z,k\right )\,\left (33192121254912\,x-\mathrm {root}\left (z^6-\frac {z^4}{1640736}+\frac {235\,z^3}{129178426752}-\frac {z^2}{3100282242048}-\frac {1}{158171241119638192128},z,k\right )\,\left (6940988288557056\,x+168897381688221696\right )+28563737812992\right )\right )\right )\right )\right )\,\mathrm {root}\left (z^6-\frac {z^4}{1640736}+\frac {235\,z^3}{129178426752}-\frac {z^2}{3100282242048}-\frac {1}{158171241119638192128},z,k\right ) \] Input:
int(x/(108*x^2 + 324*x^3 + 18*x^4 + x^6 + 216),x)
Output:
symsum(log(x + root(z^6 - z^4/1640736 + (235*z^3)/129178426752 - z^2/31002 82242048 - 1/158171241119638192128, z, k)*(216*x + root(z^6 - z^4/1640736 + (235*z^3)/129178426752 - z^2/3100282242048 - 1/158171241119638192128, z, k)*(51018336*x - root(z^6 - z^4/1640736 + (235*z^3)/129178426752 - z^2/31 00282242048 - 1/158171241119638192128, z, k)*(277947894528*x - root(z^6 - z^4/1640736 + (235*z^3)/129178426752 - z^2/3100282242048 - 1/1581712411196 38192128, z, k)*(33192121254912*x - root(z^6 - z^4/1640736 + (235*z^3)/129 178426752 - z^2/3100282242048 - 1/158171241119638192128, z, k)*(6940988288 557056*x + 168897381688221696) + 28563737812992)))))*root(z^6 - z^4/164073 6 + (235*z^3)/129178426752 - z^2/3100282242048 - 1/158171241119638192128, z, k), k, 1, 6)
\[ \int \frac {x}{216+108 x^2+324 x^3+18 x^4+x^6} \, dx=\int \frac {x}{x^{6}+18 x^{4}+324 x^{3}+108 x^{2}+216}d x \] Input:
int(x/(x^6+18*x^4+324*x^3+108*x^2+216),x)
Output:
int(x/(x**6 + 18*x**4 + 324*x**3 + 108*x**2 + 216),x)